Floquet Theory

According to Floquet, a solution for the problem of periodic linear systems can be obtained through a change of coordinates that changes that system into one with constant, real coefficients. The final result using this is

$$X(t) = P(t) e^{Rt}X_{0}\,$$

where P(t) is a T-periodic matrix and R is a constant real matrix. We can establish a relation for the values of X related to the periods of the original A matrix:

$$X(nT) = P(nT) e^{nRT}X_{0}\,$$ $$X(nT) = P(T) e^{RT^{n}}X_{0}\,$$ $$X(nT) = P(T) B^{n}X_{0}, B=e^{RT}\,$$

If $$\lambda\,$$ is an eigenvalue of R and $$\mu\,$$ an eigenvalue of B,

$$\mu=e^{\lambda t}\,$$

This result can be used, for example, at Matthieu equations for resonance in forced oscillators, or, in a more general form, to solve Hill equations.